Solving Rational Inequalities Expressing Solution In Interval Notation

Rational inequalities, a crucial topic in algebra and calculus, involve comparing a rational function (a fraction where the numerator and denominator are polynomials) to zero or another expression. Mastering the techniques to solve these inequalities is essential for understanding the behavior of functions and solving related problems. This guide provides a step-by-step approach to solving rational inequalities, along with examples and explanations to solidify your understanding.

Understanding Rational Inequalities

At its core, a rational inequality compares two rational expressions using inequality symbols such as <, >, ≤, or ≥. The general form of a rational inequality is:

f(x) / g(x) > 0, f(x) / g(x) < 0, f(x) / g(x) ≥ 0, or f(x) / g(x) ≤ 0

where f(x) and g(x) are polynomials. Solving these inequalities involves finding the set of all x values that satisfy the inequality.

Key Concepts and Definitions

Before diving into the solution process, it's crucial to understand some key concepts:

• Rational Expression: A fraction where both the numerator and denominator are polynomials.
• Critical Values: The values of x where the rational expression is either equal to zero (zeros of the numerator) or undefined (zeros of the denominator). These values are crucial because they divide the number line into intervals where the expression's sign remains constant.
• Interval Notation: A way to represent a set of numbers using intervals. For example, (a, b) represents all numbers between a and b, excluding a and b, while [a, b] includes a and b.

Step-by-Step Method to Solve Rational Inequalities

Solving rational inequalities requires a systematic approach. Here’s a detailed method:

Step 1: Rewrite the Inequality

The first step is to rewrite the inequality so that one side is zero. This involves moving all terms to one side of the inequality. For example, if you have:

f(x) / g(x) > h(x)

you would rewrite it as:

f(x) / g(x) - h(x) > 0

Then, combine the terms on the left side into a single rational expression. This often involves finding a common denominator.

For instance, let’s consider the inequality:

(x + 7) / (x - 8) ≥ 0

In this case, the inequality is already in the desired form, with zero on one side. This simplifies the process, allowing us to proceed directly to the next step.

Step 2: Find the Critical Values

Finding the critical values is a pivotal step in solving rational inequalities. Critical values are the points where the expression can change its sign. These values occur where the numerator equals zero or the denominator equals zero.

• Zeros of the Numerator: Set the numerator equal to zero and solve for x. These values make the entire rational expression equal to zero.
• Zeros of the Denominator: Set the denominator equal to zero and solve for x. These values make the rational expression undefined, as division by zero is not allowed.

Using these critical values, we divide the number line into intervals. Within each interval, the sign of the rational expression remains constant. This is because the expression can only change its sign at the critical values. The critical values act as boundaries that separate regions where the expression is either positive or negative. Understanding this behavior is essential for accurately solving the inequality.

For the inequality (x + 7) / (x - 8) ≥ 0, we identify the critical values as follows:

1. Numerator:

   x + 7 = 0
   x = -7
   

   This is a critical value because it makes the numerator zero, and thus, the entire expression zero.

2. Denominator:

   x - 8 = 0
   x = 8
   

   This is a critical value because it makes the denominator zero, rendering the expression undefined.

These critical values, -7 and 8, divide the number line into three intervals: (-∞, -7), (-7, 8), and (8, ∞). These intervals are crucial because the sign of the rational expression (x + 7) / (x - 8) remains consistent within each interval. To determine the solution of the inequality, we need to test the sign of the expression in each of these intervals.

Step 3: Create a Sign Chart

A sign chart is an invaluable tool for visualizing the intervals created by the critical values and determining the sign of the rational expression in each interval. Constructing a sign chart involves several key components that help organize the analysis.

First, draw a number line and mark the critical values on it. These values divide the number line into distinct intervals. Next, list the factors from the numerator and the denominator separately. For each factor, determine the intervals where it is positive or negative. The sign of each factor within each interval can be easily deduced by testing a value from within that interval.

To test the sign of the entire rational expression in each interval, choose a test value from within the interval and substitute it into the expression. The resulting sign (positive or negative) indicates the sign of the expression throughout that interval. This process is crucial for understanding the behavior of the rational expression across the entire number line.

The overall sign of the rational expression in each interval is determined by the product or quotient of the signs of its factors. For instance, if both the numerator and the denominator are positive in an interval, the rational expression is positive. Conversely, if the numerator is positive and the denominator is negative, the expression is negative.

Consider our example inequality, (x + 7) / (x - 8) ≥ 0. We've already identified the critical values as -7 and 8. Now, we construct the sign chart:

In this sign chart:

• The first row lists the intervals created by the critical values.
• The second and third rows show the signs of the factors (x + 7) and (x - 8) in each interval.
• The last row indicates the sign of the entire expression (x + 7) / (x - 8) in each interval, obtained by dividing the signs of the factors.

Step 4: Determine the Solution Set

The solution set is the set of all x values that satisfy the original inequality. To determine this set, refer to the sign chart and identify the intervals where the rational expression meets the inequality’s condition. It’s essential to consider whether the inequality includes equality (≤ or ≥) or strict inequality (< or >), as this affects whether the critical values are included in the solution set.

If the inequality includes equality (≤ or ≥), the zeros of the numerator are part of the solution because they make the expression equal to zero. However, the zeros of the denominator are always excluded because they make the expression undefined.

For the inequality (x + 7) / (x - 8) ≥ 0, we want to find where the expression is greater than or equal to zero. Examining the sign chart:

• The expression is positive in the intervals (-∞, -7) and (8, ∞). Therefore, these intervals are part of the solution.
• At x = -7, the expression equals zero, which satisfies the